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Trigonometry MCQ

1) Which of the following is the correct value of cot 100.cot 200.cot 600.cot 700.cot 800?

  1. 1/√3
  2. √3
  3. -1
  4. 1

Answer: (a) 1/√3

Explanation: Here, we can apply the formula -

cot A. cot B = 1 (when A + B = 900)

= (cot 200 . cot 700) x (cot 100 . cot 800) x cot 600

= 1 x 1 x 1/√3

= 1/√3

So, the correct value of cot 100.cot 200.cot 600.cot 700.cot 800 = 1/√3


2) If a sin 450 = b cosec 300, what is the value of a4/b4?

  1. 63
  2. 43
  3. 23
  4. None of the above

Answer: (b) 43

Explanation: Given a sin 450 = b cosec 300

So, a/b = cosec 300/ sin 450

a/b = 2/( 1/√2)

a/b = 2√2/1

a4/b4 = (2√2/1)4

a4/b4 = 64/1

or,

a4/b4 = 43


3) If tan θ + cot θ = 2, then what is the value of tan100 θ + cot100 θ?

  1. 1
  2. 3
  3. 2
  4. None of the above

Answer: (c) 2

Explanation: Given tan θ + cot θ = 2

Put θ = 450, above equation will satisfy as,

1 + 1 = 2

So, θ = 450,

= tan100 450 + cot100 450

= 1100 + 1100

= 2


4) If the value of α + β = 900, and α : β = 2 : 1, then what is the ratio of cos α to cos β ?

  1. 1 : 3
  2. √3 : 1
  3. 1 : √3
  4. None of the above

Answer: (c) 1 : √3

Explanation: Given α + β = 900, and α : β = 2 : 1

So, we can say that 2x + x = 900

3x = 900, which give

x = 300

So, α = 2x = 60

β = x = 30

cos α / cos β = cos 600 / cos 300

=> (1/2) / (√3/2)

or, 1/2 * 2/√3

= 1/√3

Or the ratio between cos α : cos β = 1 : √3


5) If θ is said to be an acute angle, and 7 sin2 θ + 3 cos2 θ = 4, then what is the value of tan θ?

  1. 1
  2. √3
  3. 1/√3
  4. None of the above

Answer: (c) 1/√3

Explanation: Given 7 sin2 θ + 3 cos2 θ = 4

=> 7 sin2 θ + 3 (1 - sin2 θ) = 4

=> 7 sin2 θ + 3 - 3sin2 θ = 4

Then, 4sin2 θ = 1

Or, sin θ = 1/2

So, θ = 300

Now, put θ = 300 in tan θ, we will get,

tan θ = 1/√3

Alternate

We can directly check the equation by putting values of θ. Let's put θ = 300

7 sin2 300 + 3 cos2 300= 4

Then, 7 * 1/4 + 3 * 3/4 = 4

So, 7/4 + 9/4 = 4

16/4 = 4

Or, 4 = 4 (so, it satisfy the condition)

Now, tan 300 = 1/√3


6) If tan θ - cot θ = 0, what will be the value of sin θ + cos θ?

  1. 1
  2. √2
  3. 1/√2
  4. None of the above

Answer: (b) √2

Explanation: Given tan θ - cot θ = 0

Let's put θ = 450 in order to satisfy the above equation

tan 450 - cot 450 = 0

1 - 1 = 0 (equation satisfied with θ = 450)

Now, put θ = 450 in sin θ + cos θ, we will get

= sin 450 + cos 450

= 1/√2 + 1/√2

= √2


7) If θ is said to be an acute angle, and 4 cos2 θ - 1 = 0, then what is the value of tan (θ - 150)?

  1. 1
  2. √2
  3. 1/√3
  4. None of the above

Answer: (a) 1

Explanation: Given 4 cos2 θ - 1 = 0

4 cos2 θ = 1

cos2 θ = 1/4

cos θ = 1/2

Or, θ = 600

So, tan (θ - 150) = ?

=> tan (600 - 150)

= tan 450

= 1


8) If the value of θ + φ = π/2, and sin θ = 1/2, what will be the value of sinφ?

  1. 1
  2. √2
  3. √3/2
  4. 2/√3

Answer: (c) √3/2

Explanation: Given θ + φ = π/2

It can be written as, θ + φ = 900 (as π = 1800) …….(i)

sin θ = 1/2

or, θ = 300

On putting the value of θ = 300 in equation (i), we will get,

300 + φ = 900

So, φ = 600

Then, sin φ = sin 600 = √3/2


9) What will be the value of 2cos2 θ - 1, if cos4 θ - sin4 θ = 2/3?

  1. 1
  2. 2
  3. 3/2
  4. 2/3

Answer: (d) 2/3

Explanation: Given cos4 θ - sin4 θ = 2/3

Now, here we can apply the formula -

a4 - b4 = (a2 - b2) (a2 + b2)

So, (cos2 θ - sin2 θ) (cos2 θ + sin2 θ) = 2/3

So, 1 x (cos2 θ - sin2 θ) = 2/3 (because cos2 θ + sin2 θ = 1)

=> cos2 θ - (1 - cos2 θ) = 2/3 (because sin2 θ = 1 - cos2 θ)

So, 2cos2 θ - 1 = 2/3


10) What will be the value of 1 - 2sin2 θ, if cos4 θ - sin4 θ = 2/3?

  1. 1
  2. 2
  3. 3/2
  4. 2/3

Answer: (d) 2/3

Explanation: Given cos4 θ - sin4 θ = 2/3

Now, here we can apply the formula -

a4 - b4 = (a2 - b2) (a2 + b2)

So, (cos2 θ - sin2 θ) (cos2 θ + sin2 θ) = 2/3

So, 1 x (cos2 θ - sin2 θ) = 2/3 (because cos2 θ + sin2 θ = 1)

=> (1 - sin2 θ) - sin2 θ = 2/3

So, 1 - 2sin2 θ = 2/3


11) What is the value of tan θ/(1 - cot θ) + cot θ/(1 - tan θ)?

  1. tan θ + cot θ + 1
  2. tan θ - cot θ - 1
  3. tan θ - cot θ + 1
  4. None of the above

Answer: (a) tan θ + cot θ + 1

Explanation: tan θ/(1 - (1/tan θ) + (1/tan θ)/(1 - tan θ)

= tan2 θ/ (tan θ - 1) - 1/tan θ(tan θ - 1)

= tan3 θ - 1/tan θ(tan θ - 1)

Apply the formula, a3- b3 = (a - b) (a2 + ab + b2)

= (tan θ - 1) (tan2 θ + tan θ + 1) / tan θ(tan θ - 1)

= (tan2 θ + tan θ + 1) / tan θ

On taking tan θ common from the numerator, we will get,

= tan θ + cot θ + 1


12) What is the value of sin θ/(1 + cos θ) + sin θ/(1 - cos θ), where (00 < θ < 900)?

  1. 2cosec θ
  2. 2tan θ
  3. 2cot θ
  4. None of the above

Answer: (a) 2cosec θ

Explanation: Given, sin θ/(1 + cos θ) + sin θ/(1 - cos θ)

= [sin θ (1 - cos θ) + sin θ (1 + cos θ)] / [(1 - cos θ) (1 + cos θ)]

= [sin θ - sin θ cos θ + sin θ + sin θ cos θ] / [1 - cos2 θ]

= 2 sin θ / sin2 θ

= 2 cosec θ


13) What will be the value of (√3 tanθ + 1), if r sinθ = 1, and r cosθ = √3?

  1. 2
  2. 1
  3. 0
  4. None of the above

Answer: (a) 2

Explanation: Given, r sinθ = 1, and r cosθ = √3

r sinθ / r cosθ = 1/√3

tanθ = 1/√3

or √3 tanθ = 1

So, √3 tanθ + 1= 1 + 1

= 2


14) What is the value of (tan2 θ - sec2 θ)?

  1. 2
  2. -1
  3. 1
  4. None of the above

Answer: (b) -1

Explanation: (tan2 θ - sec2 θ)

= sin2 θ/cos2 θ - 1/cos2 θ

= (sin2 θ - 1) / cos2 θ

= - cos2 θ/cos2 θ

= -1


15) If sin θ = 0.7, then what is the value of cosθ, if 00 <= θ < 900?

  1. √51
  2. √49
  3. 0.3
  4. None of the above

Answer: (a) √0.51

Explanation: Given sin θ = 0.7

As we know, sin2 θ + cos 2 θ = 1

So, (0.7)2 + cos 2 θ = 1

Then, 0.49 + cos 2 θ = 1

=> cos2 θ = 1 - 0.49

cos θ = √0.51


16) What is the value of tan3θ, If tan7θ.tan2θ = 1?

  1. √3
  2. 1/√3
  3. -1/√3
  4. None of the above

Answer: (b) 1/√3

Explanation: Given tan7θ.tan2θ = 1

As we know, if tanA . tanB = 1 then, A + B = 900

So, 7θ + 3θ = 900

=> 9θ = 900

Or, θ = 100

Now, we have to find tan3θ

So, put θ = 100 in tan3θ, we will get

tan 300 = 1/√3


17) What will be the value of 3cos800.cosec100 + 2cos590.cosec310?

  1. 3
  2. 1
  3. 5
  4. None of the above

Answer: (c) 5

Explanation: 3cos800.cosec100 + 2cos590.cosec310 = ?

According to the identity, [if A + B = 900 then, cosA.cosecB = 1]

So, 3cos800.(1/sin100) + 2cos590.(1/sin310)

= 3cos800.(1/sin(900 - 800)) + 2cos590.(1/sin(900 - 590))

=> 3cos800/cos 800) + 2cos590/cos 590 ( because sin (900 - θ) = cos θ)

= 3 + 2

= 5


18) If sin (θ + 180) = cos 600, then what is the value of cos5θ, where 00 < θ < 900?

  1. 0
  2. 1/2
  3. 1
  4. 2

Answer: (b) 1/2

Explanation: Given sin (θ + 180) = cos 600

sin (θ + 180) = cos (900 - 300)

So, sin (θ + 180) = sin300

Then, θ = 300 - 180

θ = 120

So, cos5θ = cos 5 x 120

= cos 600

= 1/2


19) If cos A = 2/3, then what is the value of tan A?

  1. 0
  2. 1/2
  3. 5/2
  4. √5/2

Answer: (d) √5/2

Explanation: According to the trigonometric identities,

1 + tan2 A = sec2 A

And we know, sec A = 1/cos A

So, sec A = 1/(2/3) = 3/2

Then, 1 + tan2 A = (3/2)2 = 9/4

=> tan2 A = 9/4 - 1

=> tan2 A = 5/4

So, tan A = √5/2


20) What will be the simplified value of (sec A sec B + tan A tan B)2 - ( sec A tan B + tan A sec B)2?

  1. 0
  2. 1
  3. -1
  4. 2

Answer: (b) 1

Explanation: The question is in the form of (a + b)2

So, on applying the identity, and after expanding the given equation, we will get -

=> sec2 A sec2 B + tan2 A tan2 B + 2 sec A sec B tan A tan B - sec2 A tan2 B - tan2 A sec2 B - 2 sec A tan B tan A sec B

=> Then, sec2 A [sec2 B - tan2 B] - tan2 A [sec2 B - tan2 B]

So, it will be written as [sec2 A - tan2 A] [sec2 B - tan2 B]

= 1 x 1

= 1.


21) What is the simplified value of (cosec A - sin A)2 + (sec A - cos A)2 - (cot A - tan A)2?

  1. 0
  2. 1
  3. -1
  4. 2

Answer: (b) 1

Explanation: The question is in the form of (a - b)2

(a - b)2 = a2 + b2 - 2ab

So, on applying the identity, and after expanding the given equation, we will get -

=> cosec2 A + sin2 A - 2 cosec A sin A + sec2 A + cos2 A - 2 sec A cos A - cot2 A - tan2 A + 2 cot A tan A

After solving it with using trigonometric identities, we will get -

=> (cosec2 A - cot2 A) + (sin2 A + cos2 A) + (sec2 A - tan2 A) -2

= 1 + 1 + 1 - 2

= 3 - 2

= 1

Alternate method

(cosec A - sin A)2 + (sec A - cos A)2 - (cot A - tan A)2

We can solve it directly by putting θ = 450

= (cosec 450 - sin 450)2 + (sec 450 - cos 450)2 - (cot 450 - tan 450)2

= (√2 - 1/√2)2 + (√2 - 1/√2)2 - (1 - 1)2

= 1/2 + 1/2 - 0

= 1


22) What will be the value of sec4 θ - tan4 θ, if sec2 θ + tan2 θ = 7/12?

  1. 1/2
  2. 7/12
  3. 1
  4. 2/3

Answer: (b) 7/12

Explanation: Given sec2 θ + tan2 θ = 7/12

Now, here we can apply the formula -

a4 - b4 = (a2 - b2) (a2 + b2)

sec4 θ - tan4 θ = (sec2 θ - tan2 θ) (sec2 θ + tan2 θ)

= 1 x (sec2 θ + tan2 θ) {because 1 + tan2 θ = sec2 θ}

= 1 x 7/12

= 7/12


23) What is the value of cos2 200 + cos2 700?

  1. √2
  2. 0
  3. 1
  4. None of the above

Answer: (c) 1

Explanation: cos2 200 + cos2 700

We can write cos2 200 as cos2 (900 - 700)

So, cos2 (900 - 700) + cos2 700

= sin2 700 + cos2 700

= 1


24) If r cosθ = √3, and r sinθ = 1, what is the value of r2 tanθ?

  1. 4/√3
  2. √3/4
  3. √3
  4. None of the above

Answer: (a) 4/√3

Explanation: Given r cosθ = √3, and r sinθ = 1

r cosθ / r sinθ = 1/√3

tanθ = tan 300

Or, θ = 300

On putting, θ = 300, we will get,

r sin 300 = 1

r x ½ = 1

or r =2

Now, r2 tanθ = ?

= (2)2 tan 300

= 4 x 1/√3

= 4/√3


25) Which of the following is the correct value of tan2 A + cot2 A - sec2 A cosec2 A, where 00 < A < 900?

  1. 4
  2. 2
  3. -2
  4. None of the above

Answer: (c) -2

Explanation: We can solve it by putting θ = 450

On putting θ = 450, we will get -

= tan2 450 + cot2 450 - sec2 450 cosec2 450

= 1 + 1 - (√2)2 x (√2)2

= 2 - 4

= -2


26) If the value of tan2 θ + tan4 θ = 1, what will be the value of cos2 θ + cos4 θ?

  1. 4
  2. 1
  3. -2
  4. -1

Answer: (b) 1

Explanation: Given, tan2 θ + tan4 θ = 1 …. (i)

From equation (i),

tan2 θ ( 1 + tan2 θ ) = 1

tan2 θ ( sec2 θ ) = 1 [As according to the trigonometric identity, sec2 θ - tan2 θ = 1]

tan2 θ = 1/ sec2 θ

tan2 θ = cos2 θ ….(ii)

Now, cos2 θ + cos4 θ = ?

=> cos2 θ + (cos2)2 θ

=> tan2 θ + (tan2)2 θ

=> tan2 θ + tan4 θ

= 1 {from equation (i)}


27) If 4sin2 θ = 3, and θ is a positive acute angle, what is the value of tan θ - cot θ/2?

  1. 4
  2. 0
  3. -2
  4. -1

Answer: (b) 0

Explanation: Given, 4 sin2 θ = 3

Or, sin2 θ = 3/4

Or, sin θ = √3/2

So, θ = 600

Now, tan θ - cot θ/2 = ?

Put θ = 600

= tan 600 - cot 600/2

= tan 600 - cot 300

= √3 - √3

= 0


28) If the value of sin A + cosec A = 2, then what is the value of sin7 A + cosec7 A?

  1. 1
  2. 0
  3. 2
  4. 3

Answer: (c) 2

Explanation: It is given that sin A + cosec A = 2 ……(i)

On putting A = 900, then above condition will satisfy

sin 900 + cosec 900 = 2

or, 1 + 1 = 2 (as the equation satisfies, so, A = 900)

Now, sin7 A + cosec7 A = ?

=> sin7 900 + cosec7 900

=> 17 + 17

= 2


29) What is the value of (2tan 300) / (1 + tan2 300)?

  1. cos 450
  2. cos 900
  3. sin 600
  4. sin 300

Answer: (c) sin 600

Explanation: tan 300 = 1/√3

(2tan 300) / (1 + tan2 300) = ?

= (2 x 1/√3) / (1 + (1/√3)2)

= (2/√3) / (4/3)

= 6/4√3

= Or √3/2, which is equal to option C, i.e., sin600.


30) What is the value of (sin 300 + cos 600) - (sin 600 + cos 300)?

  1. 1 + √2
  2. 1 + 2√2
  3. 1 + √3
  4. 1 + 2√3

Answer: (c) 1 + √3

Explanation: Let's see the values -

sin 300 = 1/2

cos 600 = 1/2

sin 600 = √3/2

cos 300 = √3/2

So, (1/2 + 1/2) - (√3/2 + √3/2)

= 1 - 2√3/2

Or, 1 - √3


31) If 1 + cos2 θ is equal to 3 sin θ.cos θ, then what is the value of cot θ?

  1. 1
  2. 0
  3. 2
  4. 3

Answer: (a) 1

Explanation: It is given that, 1 + cos2 θ = 3 cos θ.sin θ

On dividing both sides by sin2 θ, we will get

1+cos2 θ / sin2 θ = 3 cos θ.sin θ/sin2 θ

cosec2 θ + cot2 θ = 3 cot θ

=> 1 + cot2 θ + cot2 θ = 3 cot θ [because 1 + cot2 θ = cosec2 θ]

=> 1 + 2cot2 θ = 3 cot θ

=> 2 cot2 θ = 3 cot θ - 1

Let's try to put θ = 450

=> 2cot2 450 - 3 cot 450 + 1 = 0

=> 2 -3 + 1 = 0

=> 0 = 0 (satisfies)

So, θ = 450

cot θ = cot 450

= 1


32) If the value of tan θ = 4/3, then which of the following is the correct value of (3 sin θ + 2 cos θ) / (3 sin θ - 2 cos θ) =?

  1. 1
  2. -3
  3. 2
  4. 3

Answer: (d) 3

Explanation: It is given that, tan θ = 4/3

=> sin θ / cos θ = 4/3

So sin θ = 4, and cos θ = 3

Now, on putting the values of sin θ and cos θ in (3 sin θ + 2 cos θ) / (3 sin θ - 2 cos θ), we will get -

= 3x4 + 2x3/ 3x4 - 2x3

= 18/6

= 3


33) If the value of tan 150 is 2 - √3, then what is the value of tan 150 cot 750 + tan 750 cot 150?

  1. 14
  2. -13
  3. 21
  4. -14

Answer: (a) 14

Explanation: tan 150 cot 750 + tan 750 cot 150 = ?

tan 150 cot (900 - 150) + tan (900 - 150) cot 150 [as cot (900 - θ) = tan θ, and tan (900 - θ) = cot θ]

= tan2 150 + cot2 150 …..(i)

cot 150 = 1/tan 150

= 1 / 2-√3

= (1 / 2-√3) x (2+√3 / 2+√3)

So, cot 150 = 2 + √3

So, on putting the values of cot 150 and tan 150 in equation (i), we will get

= (2 - √3)2 + (2 + √3)2

= 4 + 3 - 2√3 + 4 + 3 + 2√3

= 14


34) Which of the following is the correct relation between A and B, if A = tan 110 . tan 290, and B = 2 cot 610 . cot 790?

  1. A = B
  2. A = -B
  3. A = 2B
  4. 2A = B

Answer: (d) 2A = B

Explanation: Given A = tan 110 . tan 290, and B = 2 cot 610 . cot 790

A / B = tan 110 . tan 290 / 2 cot 610 . cot 790

= [tan 110 . tan 290] / [2 cot (900 - 290) . cot (900 - 110)]

= tan 110 . tan 290 / 2 tan 110 . tan 290

= 1/2

So, A/B = 1/2

Or, 2A = B


35) If sin θ x cos θ = 1/2, then what is the value of sin θ - cos θ?

  1. 0
  2. 1
  3. -1
  4. None of the above

Answer: (a) 0

Explanation: Given sin θ x cos θ = 1/2

On multiplying both sides by 2, we will get -

2 sin θ x cos θ = 1

sin 2θ = 1 (because sin 2θ = 2 sinθ cosθ)

So, 2θ = 900

=> θ = 450

Therefore, sin θ - cos θ = ?

=> sin 450 - cos 450

= 1/√2 - 1/√2

= 0


36) If the value of sin(θ + 300) is 3/√12, then what is the value of cos2 θ?

  1. 3/4
  2. 4/3
  3. 1/4
  4. None of the above

Answer: (a) 3/4

Explanation: Given sin (θ + 300) = 3/√12

It can be written as sin (θ + 300) = 3/2√3

Or, sin (θ + 300) = √3/2

=> sin (θ + 300) = sin 600

=> θ + 300 = 600

=> θ = 300

On putting θ = 300, in cos2 θ, we will get

cos2 300 = (√3/2)2

= 3/4


37) If the value of 4 cos2θ - 4√3 cos θ + 3 = 0, then what is the value of θ?

  1. 600
  2. 300
  3. 450
  4. None of the above

Answer: (b) 300

Explanation: In this example, we can find the value of θ by putting values given in options. It is the hit and trial approach. The value that satisfies the given equation will be considered as the value of θ.

Given 4 cos2θ - 4√3 cos θ + 3 = 0

So, in option A θ = 600 is given, let's put it and see whether the equation will be satisfied or not -

4 cos2600 - 4√3 cos 600 + 3 = 0

4 x (1/2)2 - 4√3 x 1/2 + 3 = 0

=> 4/4 - 4/2√3 + 3 = 0

=> 4 - 4/2√3 = 0

=> 4(1 - 1/2√3) = 0 (will not satisfy the equation)

So, in option B, θ = 300 is given, let's put it and see whether the equation will be satisfied or not -

4 cos2300 - 4√3 cos 300 + 3 = 0

4 x (√3 /2)2 - 4√3 x √3/2 + 3 = 0

=> 3 - 6 + 3 = 0

=> 6 - 6 = 0

=> 0 = 0 (Equation satisfied)

So, option B is correct, and the value of θ is 300.


38) Which of the following is the correct value of cos2 550 + cos2 350 + sin2 650 + sin2 250?

  1. 0
  2. 3
  3. 2
  4. None of the above

Answer: (c) 2

Explanation: cos2 550 + cos2 350 + sin2 650 + sin2 250

=> cos2 (900 - 350) + cos2 350 + sin2 650 + sin2 (900 - 650)

=> (sin2 350 + cos2 350) + (sin2 650 + cos2 650)

=> 1 + 1

= 2


39) If the value of tan 90 = p/q, then what is the value of sec2 810/ 1 + cot2 810?

  1. p2/q2
  2. 1
  3. q2/p2
  4. None of the above

Answer: (c) q2/p2

Explanation: sec2 810/ 1 + cot2 810

= sec2 810/ cosec2 810

= (1/cos2 810) / (1/sin2 810)

= sin2 810 / cos2 810

= tan2 810

= tan2 (900 - 90)

= cot2 90

= q2 / p2


40) If cot 450.sec 600 = A tan 300.sin 600, then which of the following is the correct value of A?

  1. 4
  2. 1
  3. √2
  4. None of the above

Answer: (a) 4

Explanation: Given cot 450.sec 600 = A tan 300.sin 600

So, 1 x 2 = A 1/√3 x √3/2

=> 2 = A/2

So, A = 4


41) If the value of sec2 θ + tan2 θ = 7, then what is the value of θ?

  1. 00
  2. 900
  3. 600
  4. 300

Answer: (c) 600

Explanation: Given sec2 θ + tan2 θ = 7

=> 1 + tan2 θ + tan2 θ = 7

=> 2tan2 θ + 1 = 7

=> 2tan2 θ = 6

=> tan2 θ = 3

=> tan θ = √3

Or θ = 600

We can also solve this question by using the hit and trial approach. We can directly check the values of θ given in options. The value that will satisfy the given condition will be the value of θ.


42) Which of the following is the correct value of (3 / 1+tan2 θ) + 2 sin2 θ + (1 / 1+cot2 θ)?

  1. 3
  2. 9
  3. 6
  4. None of the above

Answer: (a) 3

Explanation: (3 / 1+tan2 θ) + 2 sin2 θ + (1 / 1+cot2 θ) = ?

According to the trigonometric identities, the given equation can be written as -

= 3/sec2 θ + 2 sin2 θ + 1/cosec2 θ

= 3cos2 θ + 2 sin2 θ + sin2 θ

= 3cos2 θ + 3sin2 θ

= 3(cos2 θ + sin2 θ)

= 3


43) If the value of tan2 A = 1 + 2tan2 B, then what is the value of √2 cosA - cosB?

  1. 0
  2. 9
  3. √2
  4. √3

Answer: (a) 0

Explanation: Given tan2 A = 1 + 2tan2 B

=> sec2 A - 1 = 1 + 2 (sec2 B - 1)

=> sec2 A - 1 = 1 + 2 sec2 B - 2

=> sec2 A - 1 = 2 sec2 B - 1

=> 1/cos2 A = 2/cos2 B

=> cos2 B = 2cos2 A

=> or, cos B = √2 cos A

=> So, √2 cos A - cos B = 0


44) What will be the numerical value of (4 sec2 300 + cos2 600 - tan2 450) / (sin2 300 + cos2 300)?

  1. 55/12
  2. 45/12
  3. 1/12
  4. None of the above

Answer: (a) 55/12

Explanation: Given: (4 sec2 300 + cos2 600 - tan2 450) / (sin2 300 + cos2 300)

We have to put the numerical values,

= [4 (2/√3)2 + (½)2 - (1)2] / 1

=> sec2 A - 1 = 1 + 2 (sec2 B - 1)

=> sec2 A - 1 = 1 + 2 sec2 B - 2

=> sec2 A - 1 = 2 sec2 B - 1

=> 1/cos2 A = 2/cos2 B

=> cos2 B = 2cos2 A

=> or, cos B = √2 cos A

=> So, √2 cos A - cos B = 0


45) If the value of tan A = 2, then what is the value of (cosec2 A - sec2 A) / (cosec2 A + sec2 A)?

  1. 5/3
  2. 3/5
  3. - 5/3
  4. - 3/5

Answer: (d) - 3/5

Explanation: Given: tan A = 2

(cosec2 A - sec2 A) / (cosec2 A + sec2 A) = ?

On dividing above equation with cosec2 A, we will get -

= (1 - tan2 A) / (1 + tan2 A)

= (1 - 22) / (1 + 22) [because tan A = 2]

= - 3/5


46) Which of the following is the correct value of (5/sec2 θ) + 3 sin2 θ + (2 / 1+cot2 θ)?

  1. 3
  2. 9
  3. 5
  4. None of the above

Answer: (c) 5

Explanation: (5 / sec2 θ) + 3 sin2 θ + (2 / 1+cot2 θ) = ?

According to the trigonometric identities, the given equation can be written as -

= 5cos2 θ + 3 sin2 θ + 2/cosec2 θ

= 5cos2 θ + 3 sin2 θ + 2sin2 θ

= 5cos2 θ + 5sin2 θ

= 5(cos2 θ + sin2 θ)

= 5


47) Which of the following is the correct numerical value of 5 tan2 A - 5 sec2 A + 1?

  1. - 3
  2. 9
  3. 5
  4. - 4

Answer: (d) -4

Explanation: 5 tan2 A - 5 sec2 A + 1 = ?

= 5 (tan2 A - sec2 A) + 1

= 5 ((sin2 A/cos2 A) - (1/cos2 A)) + 1

= 5((sin2 A - 1) / cos2 A) + 1

= 5(- cos2 A / cos2 A) + 1

= - 5 + 1

= - 4


48) The value of cot 300/tan 600 is -

  1. 0
  2. 9
  3. 1
  4. - 2

Answer: (c) 1

Explanation: tan 600 = √3, cot 300 = √3

So, cot 300/tan 600 = √3 / √3

= 1


49) If the value of tanP + secP = a, then what is the value of cosP?

  1. 2a/a2 + 1
  2. a2 + 1/ 2a
  3. a2 - 1/ 2a
  4. None of the above

Answer: (a) 2a/a2 + 1

Explanation: It is given that, tanP + secP = a ……(i)

As we know, the trigonometric identity, sec2 P - tan2 P = 1 {we assume θ = P}

So, we can apply the formula a2 - b2 = (a - b) (a + b)

=> (sec P - tan P) (sec P + tan P) = 1

=> (sec P - tan P) x a = 1

=> sec P - tan P = 1/a …..(ii)

So, from equation (i) and (ii), we will get -

2sec P = a + 1/a

sec P = a2+1 / 2a

So, cos P = 2a / a2+1 [as sec P = 1/cosP]


50) Suppose cos θ + sin θ = √2 cos θ, then which of the following is the correct value of cos θ - sin θ?

  1. √2 cos θ
  2. √2 sin θ
  3. -√2 cos θ
  4. -√2 sin θ

Answer: (b) √2 sin θ

Explanation: It is given that, cos θ + sin θ = √2 cos θ …..(i)

On squaring both sides, we will get,

(cos θ + sin θ)2 = (√2 cos θ)2

=> cos2 θ + sin2 θ + 2 sin θ cos θ = 2 cos2 θ

Or, 2cos2 θ - cos2 θ - sin2 θ = 2 sinθ cosθ

=> cos2 θ - sin2 θ = 2 sin θ cos θ

=> (cos θ + sin θ) (cos θ - sin θ) = 2 sin θ cos θ

=> (√2 cos θ) (cos θ - sin θ) = 2 sin θ cos θ [from equation (i)]

=> (cos θ - sin θ) = 2 sinθ cosθ / √2 cos θ

= √2 sin θ






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